Global Stability of Traveling Waves for a More General Nonlocal Reaction-Diffusion Equation

Yan, Rui; Liu, Guirong

doi:https://doi.org/10.1155/2018/6910491

Discrete Dynamics in Nature and Society

On this page

Abstract Introduction Preliminaries Applications Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2018 | Article ID 6910491 | https://doi.org/10.1155/2018/6910491

Global Stability of Traveling Waves for a More General Nonlocal Reaction-Diffusion Equation

Rui Yan¹and Guirong Liu²

Academic Editor: Pilar R. Gordoa

Received11 Feb 2018

Accepted20 May 2018

Published27 Jun 2018

Abstract

The purpose of this paper is to investigate the global stability of traveling front solutions with noncritical and critical speeds for a more general nonlocal reaction-diffusion equation with or without delay. Our analysis relies on the technical weighted energy method and Fourier transform. Moreover, we can get the rates of convergence and the effect of time-delay on the decay rates of the solutions. Furthermore, according to the stability results, the uniqueness of the traveling front solutions can be proved. Our results generalize and improve the existing results.

1. Introduction

In the study of biology and other subject fields, the reaction-diffusion equations with delays are usually utilized to depict the population distribution and physical evolution process and so forth, for instance, [1–6]. In this paper, we will study a more general reaction-diffusion equation with the initial data where is the Laplacian operator on , . Some mathematical models in the literature can be depicted by (1) with proper selections of , and . For instance, setting (1) turns to the following famous nonlocal Nicholson’s blowflies population model where is often formed as in (4). Particularly, if , (4) is Nicholson’s birth rate function. Furthermore, setting (1) turns to the classical Fisher-KPP equation as follows:

Now we impose some assumptions on (1) as follows:() is a continuous nonnegative function with and () Let , , , and for any . In addition, for any , where and .() , , and , , for any .() .

From , it is easy to see that , are two constant equilibria . Throughout this paper, a traveling front solution of (1) connecting is a nondecreasing solution with the form ; that is, it satisfies the following ordinary differential equation: where is the traveling wave speed.

In the past few years, the study on traveling waves of the reaction-diffusion equations has drawn wide attention. One of the important and difficult problems is the stability of traveling waves. For example, the authors in [7, 8] and the references therein proved the stability of traveling waves of reaction-diffusion equations without time-delay. In fact, for the time-delayed reaction-diffusion equations, Schaaf [1] firstly studied the stability of the traveling waves by applying a spectral analysis. Later, there are many great contributions on this issue on both time-delayed and nonlocal reaction-diffusion equations. For instance, in [9] the authors investigated that the traveling front solutions with noncritical speeds were globally asymptotically stable by using the super- and subsolution method. Though the study of the stability of traveling waves in monostable condition is difficult, Mei in [10] firstly showed nonlinear stability of the traveling front solutions of a time-delayed diffusive Nicholson blowflies equation by employing a technical weighted energy method. Then Mei and coauthors in [11–15] further obtained global stability using both the weighted method and the comparison principle. Among them, the authors in [11] developed and improved the wave stability results showed in [10]. By using the above methods, Wu et al. in [16] showed the exponential stability of traveling wavefronts in monostable reaction-advection-diffusion equations with nonlocal delay, which improved some previous works. In a word, there are three commonly used methods for proving the stability of traveling waves, which we mentioned above.

The most challenging problem, however, is the stability of the critical traveling wave solutions to local or nonlocal time-delayed equations. It is also very important because the critical wave speed is the spreading speed. The methods mentioned above can not be used to solve this problem. As a matter of fact, as early as in 1978, by using the maximum principle method, Uchiyama [17] gave the local stability of the traveling waves including the critical waves (no convergence rate). Immediately, Moet [18] proved that the critical waves of the KPP equation were algebraically stable by using the Green function method. Later, Kirchgässner [19] and Gallay [20] showed the stability of the critical waves by using the spectral method and the renormalization group method for parabolic equations, respectively. Recently, for some nonlocal time-delayed reaction-diffusion equations, Mei, Ou and Zhao [21] and Wang [22] proved the globally exponential stability of traveling front solutions with noncritical speeds and globally algebraical stability of traveling front solutions with critical speed by using the weighted energy method and Green’s function method. Particularly, Mei and Wang [23] considered a class of nonlocal time-delayed Fisher-KPP type reaction-diffusion equations in -dimensional space. They obtained the exponential stability of all noncritical planar wavefronts and the algebraic stability of the critical planar wavefronts by using the weighted energy method coupled with Fourier transform. Furthermore, the convergence rates were obtained in the sense with -initial perturbation. Very recently, Chern et al. [24] studied the stability of critical traveling waves for a kind of nonmonotone time-delayed reaction-diffusion equations by using the technical weighted energy method with some new developments.

The main purpose of this paper is to investigate the stability of the traveling front solutions of (1) including the traveling waves with critical speed. First, let us review the works of the existence of the traveling front solutions of (1). In [25], Wang showed the existence of traveling front solutions with speed for (1) with nonmonotone nonlinearity by constructing a closed and convex subset in a suitable Banach space and using the fixed point theorem, where is the minimal wave speed. Recently, for the reaction-diffusion equations with nonlocal delays, Tian [26] proved the existence of the traveling waves with by using the finite time-delay approximation method coupled with the monotone semiflows theorem. Then in this paper, by using the technical weighted energy method and Fourier transform, we obtained the exponential stability of traveling front solutions with noncritical speeds and the algebraic stability of the traveling front solutions with critical speed of (1). Furthermore, the convergence rates and the effect of time-delay on the decay rates of the solutions were showed. At last, motivated by Lin, Lin, and Mei [27], we show the uniqueness of traveling front solutions for (1).

The rest of the paper is organized as follows. In Section 2, we introduce some preliminaries used later. In Section 3, we will prove the stability of the traveling front solutions for (1) by using the weighted energy method and Fourier transform. According to the stability results, in Section 4 the uniqueness of the traveling front solutions can be proved. In the last section, we apply our results to some models.

2. Preliminaries

In this section, we introduce some notations as follows. We assume that represents a general constant and denotes a concrete constant. Set to be an interval, ordinarily . Take with the norm defined by Moreover, is the Sobolev space with the norm given by In addition, we assume that denotes a positive number and represents a Banach space. Also we let be the space of the -valued continuous functions on. Similarly, we can define the corresponding spaces of -valued functions on .

Next we present some previous results which will be needed in the proofs of our results later.

Lemma 1 (see [28]). Set to be the solution to the following linear time-delayed ODE with time-delay Thus where and is the delayed exponential function defined by and is the fundamental solution as

Lemma 2 (see [23]). Set and . Thus the solution of (13) satisfies where and the fundamental solution with of (17) satisfies for arbitrary constant .
Moreover, if , there is a number such that and the solution of (13) satisfies where is uniquely determined by

Moreover, clearly, conditions guarantees the existence of the traveling front solutions of (1) which was showed in [25, 26, 29]. So we have the following result.

Theorem 3 (existence of traveling waves). Suppose that hold, there is a pair determined by where For any , (1) has a nondecreasing traveling front solution such that and , while for any , (1) has no traveling front solution connecting and . Moreover, when , the traveling front solutions with noncritical speeds satisfy with some positive constant .
Furthermore, when , has two distinct positive real roots and with such that When , it holds that

Now we define a weight function with a number , with a sufficiently large number , where when , but when . Obviously for all and as .

3. The Stability of Traveling Front Solutions

In this section, we will prove the stability of all traveling front solutions with time-delay or not. Firstly, we show the following boundedness and establish the comparison principle for (1). Here we omit the proofs of these results since it is essentially the same as that of [14].

Lemma 4 (boundedness). Assume hold and the initial data satisfy then the solution of the Cauchy problem (1) and (2) exists uniquely and satisfies

Lemma 5 (comparison principle). Set and to be the solution of (1) and (2) with the initial data and , respectively. If thus

For the given , if set thus Let and be the corresponding solutions of (1) and (2) with initial data and defined in (34) respectively; that is, By Lemma 5, we can get

Now we need the following three steps.

Firstly, we will prove the convergence of to

For any , set and From (36) and (39) we get Then it can be verified that defined in (40) satisfieswith the initial data where and

By using the mollification, Zorns lemma, and energy method, we obtain the global existence for the solution of (42)-(43).

For the nonlinearity , applying Taylor’s formula to (45) and noting , we have where is some function between and ; is some function between and . Then (42) becomes Set to be the solution of the following equation with the original data : By the assumptions and the comparison principle, we see that Take then satisfies the following equation: where

When , by taking Fourier transform to (51), we have where According to Lemma 1, we get the solution of (54) as where Next by taking the inverse Fourier transform to (57), we get where the inverse Fourier transform is given by

Now we will prove the asymptotic behavior of .

Lemma 6 (decay rates for ). Suppose and . Thus, when , there is a number determined in Lemma 2 such that the solution of (51) satisfies

Proof. Set and It follows from (55) and (56) that and Because of , we can get By (62) and combining with (65)-(68), we have Let . Noticing for and , from Lemma 2, it holds that where . Then, On the other hand, by applying the characters of Fourier transform, there holds and Then, by a direct calculation, we obtain and From (71) and (75), we can get the result.

Then, for traveling front solutions with noncritical speeds, from Lemma 6 we get the following exponential decay: But if , from (25) and (27), we see that Then, for traveling front solutions with critical speeds, from Lemma 6 we get the following algebraic decay: Since for , we can directly get the following result for .

Lemma 7. There holds thatand where .

Moreover, we need to prove the decay rate of for .

Lemma 8. There holds thatwith some constant which satisfies andwhere , , and is the root of the equation .

Proof. Now we consider the following equations, for : and for , with two constants and .
When , take where is a large number. Now we can choose and such that for , , Thus, we can verify that On the other hand, for the interval , by choosing large enough, we can obtain ; that is, is an upper solution of (84). Therefore, for , we obtain When , set where is a large number. Similarly, we can obtain Therefore, for , we obtain This completes the proof.

From Lemmas 7 and 8, we prove the convergence directly as follows.

Lemma 9. There holds that
(1) if , one gets with some sufficiently small constant ;
(2) if , one gets

Secondly, we will prove the convergence of to

As shown in the above process, we can similarly prove the following convergence of to .

Lemma 10. There holds that
(1) if , one can get with some sufficiently small constant ;
(2) if , one can get

Lastly, we will prove the convergence of to

Lemma 11. There holds that:
(1) if , one can get with some sufficiently small constant ;
(2) if , one can get

Combining with the above lemmas, we can get the following stability results of the traveling front solutions for (1).

Theorem 12 (stability of traveling waves with time-delay). Suppose hold. When , for the given traveling front solution of (1) and , if the initial data holds and the initial perturbation is and , then the solution of (1) and (2) satisfies the following:
(1) if , the solution converges to the traveling wave exponentially for some constant which satisfies where and is the only root of the equation . In addition, is decreasing when and satisfies as and as ;
(2) if , the solution converges to the traveling wave algebraically

Remark 13. In Theorem 12, the assumption needed in [22] is taken away in this paper. In addition, compared to the stability results obtained in [22], the convergence rates of traveling front solutions with noncritical speeds in this paper are more accurate.

When , then (51) is reduced to By taking Fourier transform to (105), we get where It is clear that In the same way, by taking the inverse Fourier transform to (106), we have Then by a similar way of Lemma 6, we can get the following decay rates.

Lemma 14 (decay rates for ). Suppose . Thus, when , the solution of (105) satisfies

Similarly, we can prove the convergence of of to when the time-delay .

Theorem 15 (stability of traveling waves without time-delay). Suppose hold. For the given traveling wave of (1) and with speed , if the initial data holds and the initial perturbation is in , then the solution of (1) and (2) satisfies the following:
(1) if , the solution converges to the traveling wave exponentially for some constant where (2) if , the solution converges to the traveling wave algebraically

Remark 16. From Theorems 12 and 15 we conclude that the time-delay affects not only the initial perturbation, but also the convergence rates of the traveling front solutions with noncritical speeds.

4. The Uniqueness of the Traveling Front Solutions

In this part, the uniqueness of the traveling front solutions will be proved on the premise of the stability.

Theorem 17 (uniqueness of traveling front solutions). Suppose hold. Then, for the same , the traveling front solutions of (1) are unique up to translation.

Proof. For and the same , suppose that and are two different traveling front solutions. Then it follows from Theorem 3 that and where are two positive constants and is defined in Theorem 3. Let us move to with some constant . Letting , it is evident that . By choosing if , we can get Then, when , there exists such that which indicates that Next, by taking the initial data for (1) as then we get the corresponding solution to (1) From Theorem 12, for , it holds that that is, for all if . This completes the proof of the uniqueness of traveling waves up to a constant shift. Similarly, we can obtain the result when the equation is without time-delay.

5. Applications

In this section, we shall present the following results as a direct consequence of Theorems 12 and 15.

5.1. Consider the following Vector-Disease Model (See [30])

where denotes the density of infectious individual at time and site , is the recovery ratio of the infected person, and is the host-vector contact ratio. is the diffusion constant.

Clearly, (125) has two constant equilibria and . The characteristic equation at is Set There exists a corresponding number such that . In [30], the authors have justified that is the minimal wave speed of traveling wave solution connecting the two equilibria.

Set and in (1). Obviously, , for , for and where . Thus we get the following result.

Corollary 18 (stability of traveling waves). When , for the given traveling wave of (125) and , if the initial data holds and the initial perturbation is and , then the solution of (125) and (2) satisfies the following:
(1) if , the solution converges to the traveling wave exponentially for some constant which satisfies where and is the only root of the equation . In addition, is decreasing when and satisfies as . When , then and .
(2) if , the solution converges to the traveling wave algebraically

5.2. Consider the Classical Fisher-KPP Equation (7) with the Initial Data

For (7), it is easy to see that . Obviously, from Theorem 15, we obtain the following result.

Corollary 19 (stability of traveling waves). For the given traveling wave of (7) and with speed , if the initial data holds and the initial perturbation is in , then the solution of (7) and (134) satisfies the following:
(1) when , the solution converges to the traveling wave exponentially for some constant where , ;
(2) when , the solution converges to the traveling wave algebraically

Remark 20. The authors in [18–20] have showed the stability for (7). Our results generalize and develop the former results. The decay rate of the critical waves of (7) is faster than that of [19], but slower than that in [20].

5.3. Consider the following General Population Model, Which Is Evolved from a Mature Population with an Age Structure of a Single Population (See [6])

where are constants and and are Lipschitz continuous functions on any compact interval.

Let . Now we assume thatlet , for ..

Clearly, and are two constant equilibria of (139). And the characteristic equation of (139) at is Set It has proved that, for any , (139) admits a traveling front solution satisfying and . Then we can show the stability of traveling front solutions of (139).

Corollary 21. Suppose that and hold. When , for the given traveling wave of (139) and , if the initial data holds and the initial perturbation is and , then the solution of (139) and (2) satisfies the following:
(1) if , the solution converges to the traveling wave exponentially for some constant which satisfies where and is the only root of the equation . In addition, is decreasing when and satisfies as . When , then and ;
(2) if , the solution converges to the traveling wave algebraically

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All the authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (no. 11471197) and the Natural Science Foundation of Shanxi Province (no. 201601D202002).

References

K. W. Schaaf, “Asymptotic behavior and traveling wave solutions for parabolic functional-differential equations,” Transactions of the American Mathematical Society, vol. 302, no. 2, pp. 587–615, 1987.
View at: Publisher Site | Google Scholar | MathSciNet
J. Wu and X. Zou, “Traveling wave fronts of reaction-diffusion systems with delay,” Journal of Dynamics and Differential Equations, vol. 13, no. 3, pp. 651–687, 2001.
View at: Publisher Site | Google Scholar | MathSciNet
S. Ma, “Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem,” Journal of Differential Equations, vol. 171, no. 2, pp. 294–314, 2001.
View at: Publisher Site | Google Scholar | MathSciNet
J. Al-Omari and S. A. Gourley, “Monotone travelling fronts in an age-structured reaction-diffusion model of a single species,” Journal of Mathematical Biology, vol. 45, no. 4, pp. 294–312, 2002.
View at: Publisher Site | Google Scholar | MathSciNet
Z.-C. Wang, W.-T. Li, and S. Ruan, “Travelling wave fronts in reaction-diffusion systems with spatio-temporal delays,” Journal of Differential Equations, vol. 222, no. 1, pp. 185–232, 2006.
View at: Publisher Site | Google Scholar | MathSciNet
S. W. Ma, “Traveling waves for non-local delayed diffusion equations via auxiliary equations,” Journal of Differential Equations, vol. 237, no. 2, pp. 259–277, 2007.
View at: Publisher Site | Google Scholar | MathSciNet
X. Hou and Y. Li, “Local stability of traveling-wave solutions of nonlinear reaction-diffusion equations,” Discrete and Continuous Dynamical Systems - Series A, vol. 15, no. 2, pp. 681–701, 2006.
View at: Publisher Site | Google Scholar | MathSciNet
Y. Wu and X. Xing, “Stability of traveling waves with critical speeds for p-degree Fisher-type equations,” Discrete and Continuous Dynamical Systems, vol. 20, no. 4, pp. 1123–1139, 2008.
View at: Publisher Site | Google Scholar | MathSciNet
Z.-C. Wang, W.-T. Li, and S. Ruan, “Traveling fronts in monostable equations with nonlocal delayed effects,” Journal of Dynamics and Differential Equations, vol. 20, no. 3, pp. 573–607, 2008.
View at: Publisher Site | Google Scholar | MathSciNet
M. Mei, J. W. H. So, M. Y. Li, and S. S. P. Shen, “Asymptotic stability of travelling waves for Nicholson's blowflies equation with diffusion,” in Proceedings of the Royal Society of Edinburgh, Section: A, vol. 134, pp. 579–594, 2004.
View at: Publisher Site | Google Scholar | MathSciNet
C.-K. Lin and M. Mei, “On travelling wavefronts of Nicholson's blowflies equation with diffusion,” in Proceedings of the Royal Society of Edinburgh, Section A, vol. 140, pp. 135–152, 2010.
View at: Publisher Site | Google Scholar | MathSciNet
M. Mei and J. W. H. So, “Stability of strong travelling waves for a non-local time-delayed reaction-diffusion equation,” Proceedings of the Royal Society of Edinburgh, Section: A Mathematics, vol. 138, no. 3, pp. 551–568, 2008.
View at: Publisher Site | Google Scholar | MathSciNet
M. Mei, C. K. Lin, T. Lin, and J. W. H. So, “Traveling wavefronts for time-delayed reaction-diffusion equation. I. Local nonlinearity,” Journal of Differential Equations, vol. 247, pp. 495–510, 2009.
View at: Google Scholar
M. Mei, C. K. Lin, T. Lin, and J. W. H. So, “Traveling wavefronts for time-delayed reaction-diffusion equation. II. Nonlocal nonlinearity,” Journal of Differential Equations, vol. 247, pp. 511–529, 2009.
View at: Google Scholar
M. Mei, “Stability of traveling wavefronts for time-delayed reaction-diffusion equations,” in Proceedings of the 7th AIMS International Conference, Discrete and Continuous Dynamical Systems, pp. 526–535, Texas, USA, 2009.
View at: Google Scholar | MathSciNet
S.-L. Wu, W.-T. Li, and S.-Y. Liu, “Exponential stability of traveling fronts in monostable reaction-advection-diffusion equations with non-local delay,” Discrete and Continuous Dynamical Systems - Series B, vol. 17, no. 1, pp. 347–366, 2012.
View at: Publisher Site | Google Scholar | MathSciNet
K. Uchiyama, “The behavior of solutions of some nonlinear diffusion equations for large time,” Journal of Mathematics of Kyoto University, vol. 18, no. 3, pp. 453–508, 1978.
View at: Publisher Site | Google Scholar | MathSciNet
H. J. K. Moet, “A note on the asymptotic behavior of solutions of the KPP equation,” SIAM Journal on Mathematical Analysis, vol. 10, no. 4, pp. 728–732, 1979.
View at: Publisher Site | Google Scholar | MathSciNet
K. Kirchgässner, “On the nonlinear dynamics of travelling fronts,” Journal of Differential Equations, vol. 96, no. 2, pp. 256–278, 1992.
View at: Publisher Site | Google Scholar | MathSciNet
T. Gallay, “Local stability of critical fronts in nonlinear parabolic partial differential equations,” Nonlinearity, vol. 7, no. 3, pp. 741–764, 1994.
View at: Publisher Site | Google Scholar | MathSciNet
M. Mei, C. Ou, and X.-Q. Zhao, “Global stability of monostable traveling waves for nonlocal time-delayed reaction-diffusion equations,” SIAM Journal on Mathematical Analysis, vol. 42, no. 6, pp. 2762–2790, 2010.
View at: Publisher Site | Google Scholar | MathSciNet
X. H. Wang and G. Y. Lv, “Stability of traveling waves for nonlocal time-delayed reaction-diffusion equations,” Acta Mathematica Sinica Chinese Series, vol. 58, pp. 13–28, 2015.
View at: Google Scholar | MathSciNet
M. Mei and Y. Wang, “Remark on stability of traveling waves for nonlocal Fisher-KPP equations,” International Journal of Numerical Analysis and Modeling. Series B, vol. 2, no. 4, pp. 379–401, 2011.
View at: Google Scholar | MathSciNet
I.-L. Chern, M. Mei, X. Yang, and Q. Zhang, “Stability of non-monotone critical traveling waves for reaction-diffusion equations with time-delay,” Journal of Differential Equations, vol. 259, no. 4, pp. 1503–1541, 2015.
View at: Publisher Site | Google Scholar | MathSciNet
H. Wang, “On the existence of traveling waves for delayed reaction-diffusion equations,” Journal of Differential Equations, vol. 247, no. 3, pp. 887–905, 2009.
View at: Publisher Site | Google Scholar | MathSciNet
Y. Tian and P. Weng, “Asymptotic patterns of a reaction-diffusion equation with nonlinear-nonlocal functional response,” IMA Journal of Applied Mathematics, vol. 78, no. 1, pp. 70–101, 2013.
View at: Publisher Site | Google Scholar | MathSciNet
C.-K. Lin, C.-T. Lin, Y. Lin, and M. Mei, “Exponential stability of nonmonotone traveling waves for Nicholson's blowflies equation,” SIAM Journal on Mathematical Analysis, vol. 46, no. 2, pp. 1053–1084, 2014.
View at: Publisher Site | Google Scholar | MathSciNet
D. Y. Khusainov, A. F. Ivanov, and I. V. Kovarzh, “Solution of a heat equation with delay,” Nonlinear Oscillations, vol. 12, no. 2, pp. 260–282, 2009.
View at: Publisher Site | Google Scholar | MathSciNet
Z. Xu and D. Xiao, “Spreading speeds and uniqueness of traveling waves for a reaction diffusion equation with spatio-temporal delays,” Journal of Differential Equations, vol. 260, no. 1, pp. 268–303, 2016.
View at: Publisher Site | Google Scholar | MathSciNet
X.-Q. Zhao and D. Xiao, “The asymptotic speed of spread and traveling waves for a vector disease model,” Journal of Dynamics and Differential Equations, vol. 18, no. 4, pp. 1001–1019, 2006.
View at: Publisher Site | Google Scholar | MathSciNet

Copyright

Copyright © 2018 Rui Yan and Guirong Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies

Views

818

Downloads

668

Citations